Professional paper | Received: 21 -04-201 2 | Accepted: 04-1 2-201 3

The Paris Meridian Arc Length and Definition of the Metre

Miljenko SOLARIĆ and Nikola SOLARIĆ

University of Zagreb, Faculty of Geodesy, Kačićeva 26, 1 0000 Zagreb, Croatia [email protected], [email protected]

Abstract: Pierre François-André Méchain and Jean-Baptiste Joseph Delambre participated in measuring the Paris meridian arc length in order to determine the length of the metre. Jean-Baptiste Biot and François Dominique Jean Arago worked on extending the meridian at a later date.

Keywords: meridian arc length, Paris meridian, surveying, trigonometric chain, definition of length in the International System of Units, metre

1 Introduction 2 The Survey of the Paris Meridian Arc Length in Order to Define the Length of the Metre General economic growth in France during the 18th century resulted in the rapid development of trade, in- Pierre François-André Méchain (1744–1804) (Fig. 1) dustry and navigation and the need to introduce unique came from a poor family. He wanted to become an ar- units of measurement. Thus, the French Royal Academy chitect, but he was also very interested in astronomy. of Sciences started a project to establish new units of His mathematical ability was noted early. Education at measurement in 1791, as they were being demanded by L’École Nationale des Ponts et Chaussées in Paris was ex- all larger commerce networks in France and throughout pensive, so he had to leave and tutored two boys from the world. The intention was to define a new unit of len- noble families near Paris, which enabled him to pur- gth called the metre, representing one ten-millionth of chase some good astronomical instruments and pursue the meridian arc from the North Pole to the Equator. The his hobby. However, his father lost a court case and Mé- French wanted to define the metre using the meridian chain had to agree to sell his instruments in order to passing through Paris, but there were several other pro- settle the family’s debt. His instruments were bought by posals. In addition, new units of measurement were to Jérõme Lalande, who was impressed with Méchain’s be introduced in the decimal system in order to facilitate abilities. Méchain spent next years producing maps and trade. At first, the task of measuring the Paris meridian searching for comets. He produced some of maps for arc length was entrusted to Cassini IV, Legendre and Mé- military use in Germany and Italy. Afterwards, he helped chain. Subsequently, however, the Constituent Assembly Cassini IV determine the distance between the Green- of France charged François-André Méchain and Jean- wich and Paris observatories using a trigonometric net- Baptiste Joseph Delambre in 1795 with the task of sur- work. At first, he used old instruments, and later, a veying the section of the Paris meridian between Dun- repeating circle. He soon became known as the most kirk and Barcelona (URL5). Conducting the survey precise observer. In addition to comets, he recorded 29 proved to be much more dangerous than the young sci- nebulae and became a member of the Royal Academy of entists had thought, as it coincided with the difficult Sciences (URL3). times of the French Revolution. Therefore, we are going In 1791, the Committee for Weights and Measures to describe their work and the problems they faced. proposed to the National Assembly that Méchain and KiG No. 20, Vol. 1 2, 201 3 1 8 Stručni rad | Primljeno: 21 -04-201 2 | Prihvaćeno: 04-1 2-201 3

Duljina luka Pariškog meridijana i definicija metra

Miljenko SOLARIĆ i Nikola SOLARIĆ

Sveučilište u Zagrebu, Geodetski fakultet, Kačićeva 26, 1 0000 Zagreb [email protected], [email protected]

Sažetak: Na točnoj izmjeri duljine luka Pariškog meridijana za potrebe određivanja duljine metra sudjelovali su Pierre François-André Méchain i Jean-Baptiste Joseph Delambre. Poslije su na njegovu produženju radili Jean-Baptiste Biot i François Dominique Jean Arago.

Ključne riječi: duljina luka meridijana, Pariški meridijan, izmjera, trigonometrijski lanac, definicija duljine međunarodnoga sustava jedinica mjera, metar

1 . Uvod 2. Izmjera duljine luka Pariškog meridijana za potrebe definiranja duljine metra Opći ekonomski rast Francuske očitovao se u brzom usponu trgovine, industrije i pomorstva tijekom 18. sto- Pierre François-André Méchain (1744–1804) (sl. 1) ljeća i tada dolazi do potrebe uvođenja jedinstvenih potjecao je iz siromašne obitelji. Želio je postati arhitekt, mjernih jedinica. Tako je Kraljevska akademija znanosti ali imao je i posebice velik interes za astronomiju. Nje- u Francuskoj 1791. godine postavila projekt uspostave gova sposobnost za matematiku bila je odmah uočena. novih mjernih jedinica jer su to tražile sve veće trgovač- Studiranje na École Nationale des Ponts et chaussées u ke veze u Francuskoj i u svijetu. Željelo se definirati novu Parizu bilo je skupo pa je morao prekinuti studij i podu- jedinicu za duljinu nazvanu metar kao 10-milijunti dio čavati dva dječaka iz plemićkih obitelji u blizini Pariza. duljine luka meridijana od sjevernog pola do ekvatora. Tako je mogao kupiti neke astronomske instrumente Francuzi su željeli da metar bude definiran s pomoću dobre kvalitete i nastaviti svoj hobi. Međutim, kad je meridijana koji prolazi kroz Pariz, no bilo je i više drugih njegov otac izgubio parnicu, morao je prodati svoje prijedloga. Osim toga nastojalo se uvesti nove jedinice instrumente i isplatiti obiteljski dug. Njegove instru- mjera u decimalnom sustavu jer je na taj način bila olak- mente kupio je Jérõme Lalande, koji je bio impresioniran šana trgovina. Na početku su za taj zadatak izmjere Méchainovom sposobnošću. Sljedećih nekoliko godina duljine luka Pariškoga meridijana bili zaduženi Cassini izrađivao je karte i tražio komete. Neke od tih karata iz- IV., Legendre i Méchain. Međutim, poslije je Ustavotvor- radio je za vojsku u Njemačkoj i Italiji. Zatim je pomagao na skupština Francuske 1795. godine zadužila Pierrea Cassiniju IV. 1787. godine na određivanju točne udalje- François-Andréa Méchaina i Jean-Baptistea Josepha nosti između opservatorija u Greenwichu i Parizu s po- Delambrea da točno izmjere dio luka Pariškog meridija- moću trigonometrijske mreže. Na početku je mjerio na od Dunkerquea do Barcelone (URL5). Pri izvođenju te starim instrumentima, a poslije Bordinim repeticijskim izmjere pokazalo se da je to puno opasniji pothvat nego krugom. Ubrzo je stekao ugled najpreciznijeg opažača. što su mislili mladi znanstvenici. Naime, to je bilo teško Osim kometa registrirao je i 29 maglica pa je bio izabran i vrijeme kad se u Francuskoj dogodila revolucija. Zato u Kraljevsku akademiju znanosti (URL3). ćemo opisati njihov rad i probleme s kojima su se susreli. Povjerenstvo za utege i mjere predložilo je 1791. KiG Br. 20, Vol. 1 2, 201 3 1 9 SOLARIĆ, M. AND SOLARIĆ, N.: THE PARIS MERIDIAN ARC LENGTH AND DEFINITION OF THE METRE

Figure 1 . Pierre Méchain (1 744-1 804) (URL4) Figure 2. Jean-Baptiste Joseph Delambre (1 749-1 822) (URL2) Slika 1 . Pierre Méchain (1 744–1 804) (URL4) Slika 2. Jean-Baptiste Joseph Delambre (1 749–1 822) (URL2)

Delambre survey the trigonometric chain between between the latitudes of the first and last point on the Dunkirk and Barcelona, in order to determine the meridian arc was very important and required great length of the metre. The trigonometric chain was di- care. That was a reason for him to be worried. vided into two parts. Delambre was to survey the north- The situation in France was chaotic, so Méchain fol- ern section from the belfry in Dunkirk (Fig. 3) to Rodez lowed his friend’s advice and went from Spain to Genova Cathedral, a distance of 742.7 km, which had been sur- in neutral Italy, where he was informed that the meridi- veyed by Cassini de Thury prior to 1740. Méchain was to an project had been cancelled. It was relaunched in April survey the southern section from Rodez Cathedral (Fig. 1795 and Méchain was supposed to return to Paris. 4) to Montjuïc (Mont-Jouy) Fortress near Barcelona (Fig. However, he feared for his life as many scientists had 5), which had been measured less accurately as 333 km been guillotined during the French Revolution. There- in length (URL3, URL14). fore, he travelled to Marseille and Perpignan at the end MéchaintravelledtoBarcelonainJune1792,wherehe of August to resume measurement. Weather conditions set up triangulation points in Catalonia during the sum- were very poor, but Méchain continued to refuse to re- mer. He measured angles using a repeating circle in autu- turn to Paris and meet Delambre, who had finished mn, and measured the latitude by astronomical methods measuring the trigonometric network in the northern in Barcelona very carefully during the three months of section and had not measured anything for two years, as winter. Thanks to cooperation from the Spanish, he was he had been incarcerated several times during the re- allowed to survey from the top of Montjuïc Fortress. volution. Although Delambre sent Méchain all his meas- When war broke out in March 1793 between France urement data, Méchain refused to reciprocate. He was and Spain, Méchain had to leave Montjuïc (485 m above afraid that his measurement error would be discovered sea-level) due to its military purpose. After being in- if he returned to Paris. So he started altering his data, jured while testing a hydraulic pump, he returned to the taking care that deviations were minimal. upper Pyrenees. However, he was still unable to return Méchain believed that accuracy could be achieved to Montjuïc because of war, so he measured the latitude with a repeating circle. He did not understand statistics from a hotel in Barcelona. By March 1794, Méchain had based on the random error theory, which had not yet connected this two points (Montjuïc – hotel) by terrest- been developed. He was also unaware of systematic er- rial measurements (angle measurements) and discove- rors, which cannot be eliminated by repeating meas- red that the two latitudes did not match. This was very urements. He felt responsible for something which was awkward for him, considering his conscientiousness not really his fault. Refraction, which differed at the top and reputation as the most successful observer with the of the hill from that at the hotel virtually at sea level in repeating circle. A difference of 3 seconds of arc in latit- Barcelona, also influenced the results. ude corresponds to 100 m in meridian arc length (Mur- Like Delambre, Méchain measured carefully the din 2009, page 102). Unfortunately, the war meant he baseline using four platinum bars 2 toises (3.9 m) in length. was unable to return to Montjuïc to check his astrono- He set the base along a straight road near Perpignan, and mical measurements. He did not say anything about it to it was about 6000 toises long (URL14). According to Bialas anyone. Méchain knew that measuring the difference (1982, page 170), at the time, baselines were measured KiG No. 20, Vol. 1 2, 201 3 20 SOLARIĆ, M. I SOLARIĆ, N.: DULJINA LUKA PARIŠKOG MERIDIJANA I DEFINICIJA METRA

Narodnoj skupštini da Méchain i Delambre točno izmje- repeticijskim krugom. Naime, razlika od 3 kutne sekunde re trigonometrijski lanac između Dunkerquea i Barcelo- u geografskoj širini odgovara 100 m u duljini luka meridi- ne za potrebe određivanja duljine metra. Trigonome- jana (Murdin 2009, str. 102). Nažalost u dvorac Montjuïc trijski lanac bio je podijeljen na dva dijela. Sjeverni dio nije mogao otići provjeriti svoja astronomska mjerenja od zvonika u Dunkerqueu (sl. 3) do katedrale u Rodezu zbog ratnog stanja. O tome nije nikoga obavijestio. Mé- (sl. 4), dug 742,7 km, koji je već prije 1740. godine prilično chain je bio svjestan da je izmjera razlike geografske širi- točno izmjerio Cassini de Thury, trebao je precizno iz- ne prve i posljednje točke na luku meridijana vrlo važna i mjeriti Delambre. Južni dio od katedrale u Rodezu do da zahtijeva najveću pozornost pri astronomskim mjere- tvrđave Montjuïc kod Barcelone (sl. 5), koji je bio manje njima. To je bio razlog za njegovu zabrinutost. točno izmjeren, dug 333,0 km, trebao je izmjeriti Méc- U Francuskoj je bilo nesređeno stanje, te je Méchain hain (URL3, URL14). na prijedlog prijatelja iz Španjolske otišao u neutralnu Méchain je u lipnju 1792. otputovao u Barcelonu, Italiju u Genovu, gdje je čuo da se otkazuje meridijanski gdje je tijekom ljeta u Kataloniji postavio neke trigono- projekt. Projekt je ponovno započet u travnju 1795. i on metrijske točke, u jesen je mjerio kutove Bordinim repe- se trebao vratiti u Pariz. Međutim, bojao se otići u Pariz ticijskim krugom, a zimi tijekom tri mjeseca mjerio je jer je znao da su za vrijeme Francuske revolucije mnogi vrlo pažljivo astronomskim načinom geografsku širinu znanstvenici završili život na giljotini. Stoga je otputo- u Barceloni. Zahvaljujući tadašnjoj kooperativnosti Špa- vao u Marseille, a zatim krajem kolovoza u Perpignan njolaca bilo mu je dopušteno da mjeri na vrhu u dvorcu kako bi nastavio mjerenja. Vremenski uvjeti za mjerenje Montjuïc. bili su loši, a Méchain je i dalje odbijao vratiti se u Pariz Kad je izbio rat između Francuske i Španjolske u ožuj- kako bi se sreo s Delambreom, koji je izvodio izmjeru tri- ku 1793. Méchain je morao napustiti dvorac Montjuïc gonometrijske mreže na sjevernom dijelu i nije imao smješten na nadmorskoj visini od 485 m, jer je imao voj- gotovo ništa izmjereno u razdoblju od dvije godine. Na- nu namjenu. Pošto je doživio povredu pri testu hidra- ime, zbog revolucije Delambre je nekoliko puta bio za- uličke crpke, nakon dužeg oporavka vratio se izmjeri na tvaran pa nije bio u mogućnosti izvoditi mjerenja. Ipak vrhove Pireneja. Međutim, nije se mogao vratiti u dvorac je sve svoje podatke izmjere poslao Méchainu, koji mu je Montjuïc zbog rata pa je mjerio geografsku širinu iz ho- odbio poslati svoje. Bojao se da će, ako se vrati u Pariz, tela u Barceloni. Do ožujka 1794. Méchain je terestričkim biti otkrivena njegova pogrešna mjerenja. Stoga je počeo mjerenjima (mjerenjem kutova) povezao te dvije točke mijenjati svoje podatke pazeći pritom da je odstupanje (Montjuïc – hotel) i otkrio da se geografska širina iz hote- od prosjeka smanjeno. la u Barceloni i ona izmjerena u dvorcu Montjuïc ne sla- Vjerovao je da se s pomoću Bordina repeticijskoga žu. To je za njega bilo jako neugodno otkriće jer je bio kruga može postići bilo koja željena točnost. No nije ra- savjestan, a slovio je i kao najuspješniji opažač s Bordinim zumio statistiku, koja se temelji na teoriji slučajnih

Figure 5. Tower of Montjuïc (Mont- Jouy) Fortress, southern point of the meridian arc (URL8) Slika 5. Toranj u tvrđavi Montjuïc Figure 4. West front of the cathedral (Mont-Jouy), južni kraj Figure 3. Belfry in Dunkirk, northern in Rodez (URL1 5) meridijanskog luka (URL8) point of the meridian arc (URL1 4) Slika 4. Zapadni pogled na katedralu Slika 3. Zvonik u Dunkerqueu, sjeverni u Rodezu (URL1 5) kraj meridijanskog luka (URL1 4)

KiG Br. 20, Vol. 1 2, 201 3 21 SOLARIĆ, M. AND SOLARIĆ, N.: THE PARIS MERIDIAN ARC LENGTH AND DEFINITION OF THE METRE using bimetal platinum and brass measuring rods, pro- quickly became renowned in that discipline. In order to posed by Borda to substantially reduce the effect of avoid problems during the revolution, he changed his temperature on rod length. last name from de Lambre to Delambre, so that it would In the meantime, Méchain had been promised the not sound aristocratic (URL1). directorship of the Paris Observatory, which was a great At first, Delambre was interested in Greek astronomy temptation, so he returned to Paris. Soon after he was and later, also in contemporary astronomy. He read La- appointed to this high position in French science, he land’s Traité d'astronomie in 1780, and he also attended his managed to initiate a new project for extending the me- lectures at the Collège de France, where he soon impressed ridian arc to the Balearic Islands. He claimed it would in- Laland with his knowledge of astronomy, becoming his crease the accuracy of meridian arc length measure- best student. In 1786, Delambre recorded Mercury in ment. In his request to Napoleon, he emphasised the transit across the Sun, which other astronomers were un- strategic importance of the Balearic Islands and the fact able to do because it happened about 40 minutes later that his project would improve relations between Spain than the predicted time, and they had not taken into con- and France. Napoleon approved the project, so Méchain sideration deviations caused by other planets. At a meet- travelled from Paris to Spain on 26 April 1803. When he ing of the Royal Academy of Sciences, Laplace presented arrived in Barcelona, he discovered the Spanish opposed his work on planet movement deviations caused by other the project and would not provide a passport for the planets. Delambre decided to observe the movement of Balearic Islands. Therefore, he surveyed the trigono- Uranus in order to verify Laplace’s theory. The Academy metric network on the Spanish coast up to Montsia, 762 awardedhimthe1789GrandPrixforcalculatingtheexact m above sea level, 160 km from Barcelona and 145 km orbit of Uranus, and he received the award again in 1792. from Valencia. However, he was still unable to see the is- The Royal Academy of Sciences established the land of Ibiza from Montsia. He contracted malaria and Committee for Weights and Measures in 1790, consist- passed away soon after. ing of Borda, Condorcet, Laplace, Legendre and Lavoisi- In the meantime, Delambre linked the geodetic tri- er. After changing the direction of its work several times, gonometric networks of Bourges, Orléans and Sologna it published a report on 19 March 1791, recommending and returned to the north to Dunkirk in order to de- that the system be based on the metre, defined as one termine the latitude by astronomical methods and ten-millionth part of the meridian arc from the North compare it with the latitude at Montjuïc. Pole to the Equator. At first, Cassini IV, Legendre and However, when Delambre worked later on Méchain’s Méchain were charged with determining distance from data for his own base (Base du Systeme métrique), he dis- Dunkirk to Barcelona in toises (the old unit) as accurately covered that Méchain had changed his results in order to as possible. Subsequently, Delambre replaced Cassini IV makethemappearmoreaccurate.Delambredidnotwant and Legendre. Delambre was charged with surveying to diminish confidence in the metric system by publish- the northern section from Dunkirk to Rodez, and Mé- ing these facts. In fact, he considered Méchain’s results chain with the southern section from Rodez to Bar- constituted important scientific discoveries rather than celona. The task was difficult enough in peacetime, but errors, and sought to preserve Méchain’s reputation. it proved even harder because of the French Revolution and wars with neighbouring countries. Jean-Baptiste Joseph Delambre (1749–1822) (Fig. 2), In June, Delambre started looking for trigonometric was a French mathematician, astronomer and geodesist. points near Paris. He was arrested in September because At the age of 15 months, he contracted smallpox and al- his authorisation was from the king, who had also been most lost his sight. Against the odds, poor eyesight did arrested. Delambre was only released after obtaining new not stop him from becoming interested in astronomy, documents. However, he was arrested again because the and it actually gradually improved. Delambre attended authorities found his documents suspicious, and he was the Jesuit College in Amiens, studying English and Ger- accused of spying. The National Convention issued him man and continuing in the same city when the Jesuits with new papers, so he resumed his work on measuring were banned in 1764. One of his teachers encouraged him the trigonometric chain. He made some progress before to continue studying in Paris. He was awarded a scholar- winter and his return to Paris, then continued south of ship to the Collège du Plessis in Paris, where he studied Dunkirk the following spring. In December 1793, work on classical languages. However, he did not get a scholarship measuring the Paris meridian was cancelled by a decision to the University, and his parents could not afford his of the Committee for Public Security. Official functions education, so he had to give lessons to noblemen’s chil- could only be executed by people who were trustworthy dren. Although he was self-taught in mathematics, he for their republican virtues and abhorrence of kings. KiG No. 20, Vol. 1 2, 201 3 22 SOLARIĆ, M. I SOLARIĆ, N.: DULJINA LUKA PARIŠKOG MERIDIJANA I DEFINICIJA METRA pogrešaka, što do tada nije bilo razvijeno. Nije znao za astronomiju s tako slabim vidom, međutim vid mu se takozvane sustavne pogreške koje se ne mogu ukloniti postepeno poboljšavao. Pohađao je isusovački kolegij u brojem ponavljanja mjerenja. Okrivljivao je samoga sebe Amiensu studirajući engleski i njemački jezik, a kad su u za ono za što nije bio kriv. Osim toga sigurno je da je na Francuskoj 1764. isusovci zabranjeni, nastavio je studi- rezultat utjecala i refrakcija i da se ona razlikovala na vr- ranje u istom gradu. Jedan od nastavnika ohrabrio ga je hu brijega i dolje u hotelu u Barceloni, koja je približno da nastavi školovanje u Parizu. Bio je nagrađen stipendi- na razini mora. jom za Collège du Plessis u Parizu, gdje je studirao klasične Méchain je, isto kao i Delambre, pažljivo mjerio du- jezike. Poslije je izgubio stipendiju, a roditelji mu nisu ljinu baze s pomoću četiri platinske šipke, svake duge 2 mogli plaćati školovanje pa je morao davati poduke djeci toisea (3,9 m). Bazu je postavio uzduž ravne ceste kod plemića kako bi nastavio studij. Iako je bio samouk u Perpignana, a bila je duga oko 6000 toisea (URL14). Prema matematici, u tom je području ubrzo stekao ugled. U do- Bialasu (1982, str. 170), duljine baza mjerene su s pomo- ba revolucije, kako bi izbjegao probleme, promijenio je ću bimetalnih mjernih šipki izrađenih od platine i mjedi prezime iz de Lambre u Delambre da ne bi zvučalo aris- i na taj se način, prema Bordinu prijedlogu bitno reduci- tokratski (URL1). rao utjecaj temperature na duljinu mjernih šipki. Na početku se Delambre zanimao za grčku astrono- U međuvremenu Méchain je dobio obećanje da će miju, a poslije za tadašnju suvremenu astronomiju. Tako biti postavljen za ravnatelja Pariškog opservatorija, što je 1780. godine pročitao Traité d'astronomie J. Lalanda, a je bilo veliko iskušenje pa se vratio u Pariz. Ubrzo je bio pohađao je i njegova predavanja na Collège de France, te izabran na taj visoki položaj u francuskoj znanosti pa se ga impresionirao svojim znanjem iz astronomije i pos- uspio izboriti za novi projekt produživanja mjerenja tao njegov najbolji učenik. Godine 1786. Delambre je meridijanskog luka na Balearsko otočje. Tvrdio je da bi snimio prolaz planeta Merkura ispred Sunčeva diska, što se tako poboljšala točnost mjerenja duljine luka meridi- drugim astronomima nije uspjelo zbog kašnjenja Mer- jana. U svom zahtjevu Napoleonu istaknuo je i strateški kurova prolaza za oko 40 minuta. Do toga je došlo jer u značaj Balearskih otoka te da bi njegov projekt mogao proračunu tablica nisu bili uzeti u račun i poremećaji od poboljšati odnose između Španjolske i Francuske. Napo- drugih planeta. Tih dana Laplace je na sastanku Kraljev- leon je taj projekt odobrio pa je Méchain otišao iz Pariza ske akademije znanosti predstavio rad o poremećajima u Španjolsku 26. travnja 1803. Kad je stigao u Barcelonu gibanja planeta uzrokovanih drugim planetima. Delam- otkrio je da se Španjolci protive njegovu projektu pa mu bre je odlučio opažati gibanja planeta Urana kako bi nisu dali putovnicu za Balearsko otočje. Stoga je mjerio provjerio Laplaceovu teoriju. Kraljevska akademija zna- trigonometrijsku mrežu na španjolskoj obali do Mont- nosti dodijelila mu je 1789. godine nagradu Grand Prix za sia, koji se nalazi na nadmorskoj visini 762 m, a udaljen je izračun točne orbite planeta Urana, a 1792. primio je od Barcelone 160 km i od Valencije 145 km. Međutim, takvu nagradu po drugi put. imao je problem što od Montsia nije uspio vidjeti otok Kraljevska akademija znanosti osnovala je 1790. go- Ibizu. Obolio je od malarije i ubrzo umro (URL3). dine Povjerenstvo za utege i mjere u sastavu: Borda, Con- U međuvremenu je Delambre povezao geodetske dorcet, Laplace, Legendre i Lavoisier. Ono je nakon neko- trigonometrijske mreže regije Bourgesa, Orléansa i So- liko promjena smjera rada u izvještaju od 19. ožujka 1791. logne i vratio se natrag na sjever do Dunkerquea kako bi odlučilo da se sustav temelji na duljini metra definiranoj astronomskim načinom odredio geografsku širinu i us- kao 10-milijunti dio meridijanskog luka od pola do ekva- poredio je s geografskom širinom na Montjuïcu. tora. Na početku su za mjerenje po mogućnosti što točni- Poslije, kad je Delambre radio na Méchainovim po- jeg iznosa udaljenosti od Dunkerquea do Barcelone izra- dacima za svoju bazu (Base du Systeme métrique), otkrio je žene u toiseima (starim jedinicama) bili zaduženi: Cassini da je Méchain promijenio svoje rezultate očitavanja ka- IV., Legendre i Méchain. Poslije je Delambre zamijenio ko bi bili naizgled precizniji. Delambre nije želio polju- Cassinija IV. i Legendrea. Delambre je bio zadužen za ljati povjerenje u metrički sustav i javno objaviti prona- mjernje sjevernoga dijela od Dunkerquea do Rodeza, a đene činjenice. Méchainove rezultate smatrao je važnim Méchain za južni dio od Rodeza do Barcelone. Taj zadatak znanstvenim otkrićem, a ne pogreškama, te je htio sa- je bio težak i za mirnodopsko vrijeme, a pokazao se mno- čuvati njegov visoki ugled. go teži zbog Francuske revolucije i ratova sa susjedima. U lipnju je Delambre počeo najprije tražiti trigono- Jean-Baptiste Joseph Delambre (1749–1822) (sl. 2) bio metrijske točke u okolici Pariza, ali je bio uhićen u rujnu je francuski matematičar, astronom i geodet. U dobi od jer je imao odobrenje kralja, koji je također bio uhićen. 15 mjeseci razbolio se od velikih boginja i gotovo je Pušten je nakon dobivanja novih dokumenata. Među- izgubio vid. Nevjerojatno je da se počeo interesirati za tim, ponovno je bio uhićen jer su se njegovi instrumenti KiG Br. 20, Vol. 1 2, 201 3 23 SOLARIĆ, M. AND SOLARIĆ, N.: THE PARIS MERIDIAN ARC LENGTH AND DEFINITION OF THE METRE

Figure 6. The trigonometric chain known as the Delambre-Méchai Meridian used to determine the length of the metre (according to URL1 1 , basemap from Natural Earth data). Slika 6. Trigonometrijski lanac nazvan Delambreov i Méchainov meridijan s pomoću kojeg je određena duljina metra (prema URL1 1 , temeljna karta iz podataka Natural Earth).

KiG No. 20, Vol. 1 2, 201 3 24 SOLARIĆ, M. I SOLARIĆ, N.: DULJINA LUKA PARIŠKOG MERIDIJANA I DEFINICIJA METRA

Table 1 . Results of the Paris meridian survey according to Méchain and Delambre, 1 792–1 798 (Bialas, 1 982, page 1 71 ) (t – toise) Tablica 1 . Rezultati izmjere Pariškog meridijana prema izmjeri Méchaina i Delambrea u razdoblju 1 792–1 798. (Bialas, 1 982, str. 1 71 ) (t – toise)

Arc between Diff. in lat. Average lat. Dist. of arc Dist. of arc for 1° Luk između Razlika Srednja Duljina Duljina luka geo. širina geo. širina mer. luka jednog stupnja

I Dunkirk (Dunkerque) – Pantheon (Paris) 2° 11’ 20.7” 49° 56' 30'' 124 945.18 t 57 076 t II Pantheon (Paris) – Evaux 2° 40’ 07.3” 47° 30' 46'' 152 291.48 t 57 066 t III Evaux – Carcassonne 2° 57’ 48.1” 44° 41' 48'' 168 849.10 t 56 979 t IV Carcassonne – Montjuïc 1° 51’ 09.6” 42° 17' 20'' 105 498.96 t 56 945 t V Dunkirk (Dunkerque) – Montjuïc 9° 40’ 25.6” 46° 11' 58'' 551 584.72 t 57 018 t tadašnjim vlastima činili sumnjivim te su ga optužili da luka uzeta baza kod Meluna, a južnog dijela trigonome- ih koristi za špijunažu. Tada je dobio potvrdu od Naci- trijske mreže baza kod Perpignana. Pritom su obje baze onalne konvencije pa je nastavio s radom na izmjeri tri- bile istovrijedne. Kod ponovljenog računanja prema De- gonometrijskog lanca. Ostvario je mali napredak do lambreu pojavile su se ipak male korekture kutova u zime i povratka u Pariz, a sljedećeg je proljeća nastavio s trokutima od samo 0,1'' (Bialas 1982, str. 171). radom južno od Dunkerquea. U prosincu 1793. radovi na Naglasimo da u to doba još nije bila poznata metoda izmjeri Pariškog meridijana prekinuti su odlukom Od- izjednačenja rezultata mjerenja s prekobrojnim mjere- bora za javnu sigurnost. Službene funkcije mogle su njima. Naime, Carl Friedrich Gauss (1777–1855) objavio obavljati samo osobe koje su pouzdane, imaju republikanske je 1809. godine knjigu Theoria motus corporum celestium in vrline i gnušaju se kraljeva. sectionibus conicis solem ambientium u kojoj je naveo da je U svibnju 1795. Delambre je nastavio s radovima koji on izmislio metodu najmanjih kvadrata i da je upotreb- su bili naglo prekinuti prije osamnaest mjeseci. Mjerio je ljava od 1795. godine. Međutim, nastao je spor s Adrien- trigonometrijsku mrežu između Orleansa i Bourgesa u Marie Legendreom (1752–1833), koji je 1805. godine drugoj polovici 1795. godine, između Bourgesa i Evauxa objavio novu metodu za određivanje orbita kometa i u u ljetu 1796. i između Evauxa i Rodeza 1797. Između tih dodatku opisao novu metodu najmanjih kvadrata, koja izmjera trigonometrijske mreže putovao je u Dunkerque igra ključnu ulogu u sređivanju podataka mjerenja u prosincu 1795. i proveo ondje nekoliko prvih mjeseci (URL6). Danas se uglavnom uzima da je C. F. Gauss tvorac 1796. godine računajući vrlo pažljivo geografsku širinu metode izjednačenja mjerenih veličina s najmanjom su- (URL1). Također, bila je tražena i visoka točnost mjerene mom kvadrata popravaka. baze tako da bi mjerilo trigonometrijske mreže moglo Primjena Bordina repeticijskoga kruga u izmjeri ku- biti što točnije utvrđeno. Vrlo točna mjerenja duljine tova donijela je također promjenu u računanju trigono- baze pokraj Meluna nedaleko Pariza obavio je Delambre metrijske mreže. Naime, kad su se mjerili kutovi s u travnju 1798. Baza je bila postavljena uzduž ceste duge kvadrantima s mogućnošću mjerne nesigurnosti od 15" oko 12 km, a mjerio ju je šest tjedana na isti način kao i (kutnih sekundi) nije trebalo računati sferni eksces, te se Méchain. Pritom se služio, gdje je to bilo moguće, Cassi- moglo računati trokute kao da su u ravnini. Međutim, nijevim trigonometrijskim točkama iz 1744. godine kad se počelo mjeriti kutove s pomoću Bordina repeti- (URL14). cijskoga kruga, kojim se mjerilo s mjernom nesigurno- Mjerenja su bila ukupno vrlo dobre kvalitete, što je šću od 1", tada se više nije moglo računati trokute u pokazala i usporedba između izravno izvedenog mjere- ravnini, već se moralo računati sa sfernim trokutima nja kontrolnih baza i računanja izvedenih preko izmje- (sferni eksces je prvi put računan u Peruanskoj ekspedi- renih kutova u trokutima. Tako je nakon računanja, ciji 1735–1743) (Solarić M. i Solarić N. 2013). Tako se mo- polazeći od glavne baze kod Meluna, preko više od 53 rao za veće trokute računati sferni eksces, tj. odstupanje trokuta, dobivena razlika između izračunane i izravno zbroja kutova u sfernom trokutu od 180°. Sferni ekscesi izmjerene duljine iznosila samo 0,16 toisea (oko 0,31 m). bili su u granicama od 0" do 2", a samo u dva slučaja kod Iz toga slijedi da je u izmjeri meridijanskog luka postig- velikih trokuta bili su 4" i 5" (Vykutil 1982, str. 422). nuta relativna točnost od 1: 37 500 (Bialas 1982, str. 171). Trigonometrijski lanac Pariškog meridijana od Dun- Taj rezultat bio je osiguran tako da je za računanje sje- kerquea do Barcelone, koji su izmjerili Méchain i Delam- vernog dijela trigonometrijske mreže meridijanskog bre od 1792. do 1799. godine, prikazan je na sl. 6. KiG Br. 20, Vol. 1 2, 201 3 25 SOLARIĆ, M. AND SOLARIĆ, N.: THE PARIS MERIDIAN ARC LENGTH AND DEFINITION OF THE METRE

In May 1795, Delambre continued the work which calculate triangles in a plane; spherical triangles had to had been cancelled abruptly eighteen months earlier. be used (spherical excess was calculated for the first He measured the trigonometric network between Or- time in expedition to Peru 1735–1743) (Solarić, M. and leans and Bourges in the second half of 1795, between Solarić N. 2013). Spherical excess, i.e. deviation of the Bourges and Evaux in the summer of 1796, and between sum of angles in a spherical triangle from 180° had to be Evaux and Rodez in 1797. Between surveys, he travelled calculated for larger triangles. Spherical excesses to Dunkirk in December 1795 and spent the first few ranged from 0" to 2", and 4" and 5" in only two cases of months of 1796 calculating the latitude very carefully large triangles (Vykutil 1982, page 422). (URL 1). A high degree of accuracy of the measured base The trigonometric chain of the Paris meridian from was also required in order for the trigonometric net- Dunkirk to Barcelona measured by Méchain and Delam- work scale to be determined as accurately as possible. bre between 1792 and 1799 is represented in Fig. 6. Latit- Delambre conducted measurements of the baseline at udes were not measured only at the starting and ending Melun near Paris in April 1798. The baseline of 12 km points but also at intermediate points at the Panthéon was set along a road, and Delambre measured it the (Paris), Evaux and Carcassonne. Measurement of the same way as Méchain. Where possible, he used Cassini’s azimuths of some sides was also done. The total length old trigonometric points from 1744 (URL14). of the Dunkirk – Montjuïc meridian arc (551 584.72 In general, the measurements were very good, toises) is also included in Table 1. which was demonstrated by comparing direct measure- Delambre informed the International Committee for ments of control bases and calculations of measured Weights and Measures of the results of the survey in angles in triangles. Thus, after calculations starting February 1799. He published details of the project in Base from the Melun main base over 53 triangles were com- du système métrique (URL12). The first of three volumes pleted, the difference between the calculated and dir- was published in 1806, containing the history of meas- ectly measured lengths was only 0.16 toise (about 0.31 uring the Earth and data on the trigonometric network m), giving a relative accuracy of 1: 37 500 (Bialas 1982, p. project. The second volume was published in 1807, con- 171). This result was ensured by taking the Melun base taining data on the accurate determination of latitudes for calculating the northern section of the trigonomet- in Dunkirk and Barcelona. The third volume was pub- ric network of the meridian arc, and the Perpignan base lished in 1810, discussing errors in calculating the for the southern section. Both bases were equally im- Earth’s eccentricity. portant. Small corrections of angles in triangles amoun- In June 1798, at the request of the National Institute, ting to only 0.1" did occur in a repeated calculation the French bishop and statesman Charles-Maurice Tal- according to Delambre (Bialas 1982, page 171). leyrand (1754–1838) invited allied and neutral nations Itshouldbenotedthatthemethodofadjustingresults to collaborate on determining final standards of basic using repeated measurements was unknown at the time. units for measuring length and other quantities. An in- Namely, Carl Friedrich Gauss (1777–1855) published his ternational committee was formed, consisting of the Theoria motus corporum celestium in sectionibus conicis solem Dutchman Van Swiden, the Swiss Tralles, the Spaniard ambientium in 1809, stating he had devised the least Ciscar and the French Laplace, Legendre, Méchain and squares method and had been using it since 1795. Delambre (Bialas 1982, page 172). The committee was to However, there was a dispute with Adrien-Marie Le- calculate the length of the meridian from the North Pole gendre (1752–1833),who in 1805 publisheda newmethod to the Equator and thus determine the length of the for determining comet orbit, describing a new least metre as one ten-millionth part of its length. Namely, squares method, which played a key role in adjusting the length of the section of the meridian arc from measurement data (URL6). Nowadays, C. F. Gauss is usu- Dunkirk to Barcelona is only about a tenth of the me- ally considered the author of the least squares method. ridian arc length from the North Pole to the Equator, ap- Application of the repeating circle in measuring proximately at the mean latitude 46° 12’. Therefore, the angles also changed the way the trigonometric network committee first had to calculate the flattening of the was calculated. Namely, when angles were measured Earth at the pole based on: with quadrants with the possibility of measuring with . the results of surveying a section of the Paris meridi- margin of error of 15" (arc-seconds), it was not neces- an arc from Dunkirk to Barcelona 9° 40' 24" in size, sary to calculate the spherical excess and triangles could surveyed by Delambre and Méchain (Table 1), and be calculated like in a plane. However, when the repeat- . the accurately surveyed meridian arc in South Ame- ing circle was employed to measure angles and its meas- rica (Peru) 3° 07' 01" in size, surveyed by Bouguer, La uring uncertainty was 1", it was no longer possible to Condamine and Godin. KiG No. 20, Vol. 1 2, 201 3 26 SOLARIĆ, M. I SOLARIĆ, N.: DULJINA LUKA PARIŠKOG MERIDIJANA I DEFINICIJA METRA

Južnoj Americi (u Peruu), dugog 3° 07' 01", koji su iz- mjerili Bouguer, La Condamine i Godin. Dobili su da je spljoštenost Zemlje jednaka f = 1:334. Iz te izračunane vrijednosti Zemljine spljoštenosti i vrijednostima izvedenih iz mjerenja na meridijanskom luku od Dunkerquea (φ = 51° 02' 09,2") do Barcelone (φ = 41° 21' 44,96") dobili su da duljina tog dijela meridijan- skog luka iznosi 551 583,6 akademijinih toisea. Do ukupne duljine meridijanskog luka od pola do Figure 7. Diagram of the dimensions of the 1 799 "final ekvatora došlo se preko polinomskog prikaza za duljinu metre", called the Archive Metre (Brezinšćak 1 971 ) kvadranta elipse uzimajući u račun i kvadratne članove Slika 7. Grafički prikaz dimenzija "konačnog metra", spljoštenosti elipsoida te su dobili da je tražena duljina nazvanog Arhivski metar, iz 1 799. godine (Brezinšćak 1 971 ) luka meridijana od pola do ekvatora jednaka 5 130 740 akademijinih toisea. Geografske širine nisu bile izmjerene samo u počet- Iz toga je slijedilo da je metar prema definiciji kao 10- noj i krajnjoj točki nego i u međutočkama Pantheon (Pa- milijunti dio luka meridijana od pola do ekvatora dug riz), Evaux i Carcassonne. Osim toga izvedena su i 0,513074 akademijina toisea, tj. da je 1 akademijin toise = odgovarajuća mjerenja azimuta nekih strana. U tablicu 1,949037 metara. 1. uvrštena je i ukupna duljina meridijanskog luka Dun- To znači da je konačna vrijednost metra kraća od kerque – Montjuïc od 551 584,72 toisea. privremenoga1 za samo 0,144 akademijine linije (1 toise = Delambre je u veljači 1799. izvijestio Međunarodno 6 stopa, 1 stopa = 144 linije). povjerenstvo za utege i mjere o rezultatima izmjere u Izrađeno je više primjeraka motki metra od platine projektu. Detalje o čitavom projektu naknadno je obja- (sl. 7), i to uz veliku pozornost s pomoću Lenoirova kom- vio u radu Base du système métrique (URL12). Prvi od tri paratora za definitivni standardni metar. Pritom su volumena objavljen je 1806. godine, a sadrži povijest motke metra imale poprečni presjek 25 mm × 4,05 mm, a mjerenja Zemlje i podatke o projektu trigonometrijske krajevi su bili točno udaljeni 1 metar pri temperaturi ta- mreže. Drugi volumen objavljen je 1807. godine i sadrži ljenja leda. Pogreška mjerenja duljine pri uspoređivanju podatke o točnim određivanjima geografskih širina u krajnjih metarskih mjera tada je iznosila oko 1/100 mili- Dunkerqueu i Barceloni. Treći volumen objavljen je metra. Kada je konačni rezultat bio poznat, odabrana su 1810., a u njemu su razmatrane pogreške u računanju dva primjerka metra čije su duljine bile najbliže defini- Zemljina ekscentriciteta. ciji metra. Jedan primjerak metra i kilograma predani su Biskup i državnik Charles-Maurice Talleyrand na čuvanje u Arhiv Francuske Republike 22. lipnja 1799., (1754–1838) je na zahtjev Nacionalnog instituta u lipnju a drugi primjerak metra i kilograma predani su na čuva- 1798. pozvao savezničke i neutralne narode da sudjeluju nje u Bureau des Longitudes u Zvjezdarnicu (Brezinšćak u utvrđivanju konačnih standarda osnovnih jedinica za 1971, str. 184). Primjerak koji se čuva u Nacionalnom ar- mjerenje duljina i ostalih veličina. Tako je bilo izabrano hivu poznat je kao Arhivski metar (Mètre des Arhives). šire međunarodno povjerenstvo u sastavu kojeg su bili: Delambre je primljen u Bureau des Longitudes 1795. Nizozemac Van Swiden, Švicarac Tralles, Španjolac Cis- godine, a postao predsjednik 1800. Za tajnika Akademije car te Francuzi: Laplace, Legendre, Méchain i Delambre znanosti postavljen je 1801. godine, što ga je činilo naj- (Bialas 1982, str. 172). Ono je imalo zadatak izračunati moćnijom osobom u francuskoj znanosti toga doba duljinu meridijana od pola do ekvatora i na taj način (URL1). Godine 1803. počeo je pobolijevati od reumatske odrediti duljinu metra kao 10-milijunti dio njegove du- groznice, ali je i dalje postizao uspjehe. Napoleon je 1809. ljine. Naime, duljina dijela meridijanskog luka od Dun- tražio od Akademije znanosti da predloži nekoga za na- kerquea do Barcelone je samo oko desetine duljine luka gradu za najbolji znanstveni rad. Nagrada je otišla De- meridijana od pola do ekvatora, drugim riječima samo lambreu za njegov rad na meridijanu. U posljednjem njezin deseti dio i to na približno srednjoj geografskoj razdoblju karijere bio je zainteresiran za povijest mate- širini 46° 12'. Zato je povjerenstvo moralo najprije izra- matike i astronomije pa je na tu temu napisao više knjiga. čunati spljoštenost Zemlje na osnovi: 1 Narodna skupština nastojala je izbjeći stari naziv za duljinu Pi- . rezultata izmjere dijela luka Pariškog meridijana od ed du Roy jer je nova vlast željela što prije zaboraviti na kralja. Dunkerquea do Barcelone, dugog 9° 40' 24", koju su Zato je Lenoir izradio privremeni metar od mjedi 1795. godi- obavili Delambre i Méchain (danih u tablici 1) ne, dug 443,44 akademijine linije. Ta duljina je određena iz prethodnih meridijanskih mjerenja: u Peruu, u Laplandu i du- . posebno točno izmjerenoga meridijanskog luka u ljine Dunkerque-Perpignan od Cassinija (URL16). KiG Br. 20, Vol. 1 2, 201 3 27 SOLARIĆ, M. AND SOLARIĆ, N.: THE PARIS MERIDIAN ARC LENGTH AND DEFINITION OF THE METRE

The calculated flattening of the Earth was f = 1:334. 3 Extending the Survey of the Paris Meridian Based on that value and values from surveys of the as far as the Balearic Islands meridian arc from Dunkirk (φ = 51° 02' 09.2") to Bar- celona (φ = 41° 21' 44.96"), the length of this section of Méchain proposed extending the survey of the Paris the meridian arc was calculated as 551 583.6 toises. meridian as far as the Balearic Islands. Unfortunately, he The total meridian arc length from the North Pole to died soon afterwards, so Jean-Baptiste Biot and François the Equator was obtained using a polynomial repres- Dominique Jean Arago continued the work in difficult entation for the length of the ellipse quadrant, taking conditions. into consideration second order members of ellipsoid flattening, resulting in a meridian arc length from the Jean-Baptiste Biot (1774–1862) (Fig. 8) was a French North Pole to the Equator equal to 5,130,740 toises. Con- physicist, astronomer and mathematician (URL9, URL10). sequently, according to the definition of the metre as He was educated at the college of Louis-le-Grand in Paris one ten-millionth part of the meridian arc from the and joined the French army in 1792. He was arrested for North Pole to the Equator, it was the equivalent of desertion. Biot continued studying mathematics and be- 0.513074 toises, i.e. 1 toise = 1.949037 m. This means the fi- came a professor of mathematics in Beauvais in 1797, and nal value of the metre was shorter than the provisional a professor of mathematical physics at the Collège de metre1 by only 0.144 lignes (1 toise = 6 pieds, 1 pied = 144 France around 1800. He became a member of the Academy lignes). of Sciences in 1803 and an external member of the Royal Several platinum metre rods (Fig. 7) were produced Swedish Academy in 1816. Biot was a proponent of New- with special care using the Lenoir comparator for the ton’s theory of gravity and he made many scientific con- definitive standard metre. The rods had a transverse tributions to optics, magnetism and astronomy. section of 25 mm × 4.05 mm, and their ends were exactly When Biot and Arago were at the Paris Observatory, 1 metre apart at the temperature of melting ice. The they discussed the termination of work on the Paris me- length measurement error in comparing end measures ridian caused by Méchain’s death and told Pierre-Simon was about 1/100 of a millimetre. When the final result Laplace (1749–1827) of their plan to extend the Paris was obtained, the two rods closest to the definition of meridian southwards to the Balearic Islands. He accep- the metre were selected. A copy of the metre and the ted their proposal and supported it in the Committee for kilogram were transferred to the Archive of the French Weight and Measures of the Academy of Sciences. Republic on June 22, 1799, and another copy of each was However, it turned out to be a much more hazardous ad- transferred to the Bureau des Longitudes in the Obser- venture than the young scientists had thought. vatory (Brezinšćak 1971, page 184). The copy preserved Biot and Arago travelled together from Paris to Spain in the National Archive was known as the Archive Metre in 1806 and started extending the trigonometric net- (Mètre des Arhives). work in the mountains of Spain. Conditions were very Finally, Delambre was admitted to the Bureau des difficult, because they had to traverse rough terrain and Longitudes in 1795 and became its president in 1800. He endure bad weather conditions, along with other prob- became the secretary of the Academy of Sciences in lems in a foreign land during wartime. Biot returned to 1801, making him the most powerful person in French Paris in 1808, and Arago remained, completing the work science at the time (URL1). In 1803, he contracted under great hardship until 1809. rheumatic fever, but continued to enjoy success. In 1809, Napoleon requested the Academy of Sciences to pro- François Dominique Jean Arago (1786–1853) (Fig. 9) pose a suitable recipient for an award for the best sci- was a French mathematician, physicist, astronomer, entific work. Delambre was chosen for his work on the geodesist and politician (URL7). He was born near Per- meridian. At the end of his career, he became interested pignan. At first, he was interested in a military career, in the history of mathematics and astronomy and wrote but later turned to mathematics. He enrolled in the Ėcole several books on those topics. Polytechnique in Paris in 1803 and became friends with Professor Denis Poisson. 1 The National Assembly wanted to avoid the old measure- When Méchain died in Spain in September 1804, ment of length Pied du Roy because the new government wanted to forget the king as soon as possible. Therefore, Denis Poisson suggested to the director of the Paris Ob- Lenoire produced a provisional metre made of brass, equ- servatory that Arago be appointed to Méchain’s post. He ivalent to 443.44 lignes in 1795. The length had been deter- used a telescope to observe the positions of stars and mined in previous meridian surveys in Peru and Lapland and by Cassini's Dunkirk-Perpignan measurements (URL also started working with Biot on the refraction of light 16). through gases. It was a very important issue in practical KiG No. 20, Vol. 1 2, 201 3 28 SOLARIĆ, M. I SOLARIĆ, N.: DULJINA LUKA PARIŠKOG MERIDIJANA I DEFINICIJA METRA

Figure 8. Jean-Baptiste Biot (1 774–1 862) (URL9) Figure 9. François Dominique Jean Arago (1 786–1 853) (URL7) Slika 8. Jean-Baptiste Biot (1 774–1 862) (URL9) Slika 9. François Dominique Jean Arago (1 786–1 853) (URL7)

3. Produživanje izmjere duljine Pariškog ratnim vremenima. Biot se vratio u Pariz iz Španjolske meridijana do Balearskog otočja 1808. godine, a Arago je ostao završiti radove uz mnogo- brojne poteškoće do 1809. godine. Prijedlog za produživanje izmjere duljine Pariškog meridijana do Balearskog otočja dao je Méchain. Naža- François Dominique Jean Arago (1786–1853) (sl. 9) lost on je ubrzo umro, a radove u teškim uvjetima nasta- bio je francuski matematičar, fizičar, astronom, geodet i vili su Jean-Baptiste Biot i François Dominique Jean političar (URL7). Rođen je u malom mjestu pokraj Per- Arago. pignana. Najprije je pokazao interes za vojnu karijeru, a poslije za matematiku. U Ėcole Polytechnique u Parizu Jean-Baptiste Biot (1774–1862) (sl. 8) bio je francuski ušao je 1803. godine i sprijateljio se s profesorom Deni- fizičar, astronom i matematičar (URL9, URL10). Studirao som Poissonom. je u College Louis-le-grand u Parizu, a 1792. pridružio se Kad je Méchain umro u Španjolskoj u rujnu 1804. Po- topništvu u francuskoj vojsci. Kad je napustio vojsku isson je predložio ravnatelju Pariškog opservatorija da na uhićen je kao dezerter. Nastavio je studij matematike i Méchainovo mjesto uzme Aragoa. On je teleskopom opa- 1797. imenovan je za profesora matematike u Beauvaisu, žao položaje zvijezda, a počeo je raditi s Biotom na re- a oko 1800. godine za profesora matematičke fizike na frakciji svjetlosti kroz plinove. To je vrlo važno pitanje u CollègedeFrance. U Akademiju znanosti izabran je 1803., a praktičnoj astronomiji, jer su vidljivi položaji zvijezda za vanjskoga člana Kraljevske švedske akademije iza- pomaknuti zbog poremećaja pri prolazu kroz atmosferu. bran je 1816. godine. Biot je bio vatreni štovatelj Newto- Nakon što je Akademijino povjerenstvo za utege i nove teorije gravitacije i dao je mnogobrojne doprinose mjere prihvatilo njihov prijedlog o produživanju Pari- znanosti u optici, magnetizmu i astronomiji. škog meridijana, Arago i Biot postali su tim koji se pri- Kad su Biot i Arago radili na Pariškom opservatoriju premao osamnaest mjeseci za savladavanje potrebnih dogovarali su se o završetku rada na Pariškom meridija- tehnika mjerenja. Rečeno im je da osim izravnog mjere- nu, koji je bio prekinut Méchainovom smrću. Plan o nja na trigonometrijskom lancu za određivanje duljine kompletiranju Pariškoga meridijana na jug do Balear- luka jednog stupnja meridijana trebaju odrediti i duljinu skih otoka priopćili su Pierre-Simonu Laplaceu sekundnog njihala na više mjesta, što bi im pomoglo u (1749–1827). On je prihvatio njihov prijedlog i podupro određivanju Zemljina gravitacijskog polja, tj. njezina ga u Povjerenstvu za utege i mjere Akademije znanosti. oblika (URL 12). Iz Pariza su otputovali 1806. godine i po- Međutim, poslije se pokazalo da je to puno opasniji pot- čeli s radom u planinama Španjolske. Pridružila su im se hvat nego što su mislili mladi znanstvenici. i dva Španjolca, Chaix i Rodriguez, koji su im bili potreb- Biot i Arago su zajedno otputovali iz Pariza 1806. i ni za rad u stranoj zemlji. počeli raditi na produživanju trigonometrijske mreže u Radeći dalje prema jugu, Arago i Biot postavili su tri- planinama Španjolske. Uvjeti rada bili su vrlo teški jer su gonometrijsku točku na vrh Mongo blizu Denija (sl. 10), morali fizički savladavati teren i vremenske nepogode, odakle su mogli vidjeti točku La Mola na malom otočiću te se susretati s raznim poteškoćama u stranoj zemlji u Formentera, udaljenom oko 100 km od vrha Mongo. KiG Br. 20, Vol. 1 2, 201 3 29 SOLARIĆ, M. AND SOLARIĆ, N.: THE PARIS MERIDIAN ARC LENGTH AND DEFINITION OF THE METRE astronomy because the visible positions of stars shifted network (Fig. 10). He also measured the latitudes at due to disturbances in the atmosphere. some points (Murdin 2009, p. 116). After the Committee for Weights and Measures ac- As the French had ambitions regarding the Balearic cepted Biot and Arago’s proposal to extend the Paris Islands, the local population interpreted Arago’s setting meridian, they became a team which took 18 months to lights the top of Galatzó hill (Catalan Mola de l'Esclop) as a prepare for conducting the necessary measurements. In sign for the invasion of enemy forces, though he was addition to direct measurements along the trigonomet- only trying to improve the visibility of the point in the ric chain, they also had to determine the length of trigonometric network. He was arrested in June 1808, second pendulum in several places, in order to facilitate but he escaped from the island in a fishing boat on 28 Ju- a determination of the Earth’s gravitational field, i.e. its ly and arrived in Algeria on 3 August. He was granted shape (URL12). In 1806, they left Paris and began work in permission to travel to Marseille, but as the ship ap- mountains of Spain. They were joined by two Spaniards, proached the port on 16 August, Arago and the crew Chaix and Rodriguez, who were indispensible to the for- were taken prisoners by the Spanish and sent to Roses eigners. on the Spanish coast, from where they were transferred Working their way southwards, Arago and Biot set to Palamos. Somehow Arago managed to board a ship up a trigonometric point to Mongo peak near Denia (Fig. bound for Marseille on 28 November 1808. 10), from where they could see La Mola on the small is- Fate played with Arago once again and bad weather land of Formentera, about 100 km from Mongo. In addi- steered the ship to the coast of North Africa, where he tion, they were able to spot Camp Vey on Ibiza. Galatzo was imprisoned by Muslims. Thanks to his resourceful- (S'Esclop) on Majorca could be determined from those ness, he reached Algeria on 25 December, where the two points. The side of the triangle from Desierto de las French consul helped him set sail for Marseille for the Palmas to Camp Vey, which is more than 150 km long, third time. This time he reached his destination without was also completed (Murdin 2009, p. 113). Triangles with any trouble on 2 July 1809. sides as long as these made it difficult to measure angles, Upon arrival in Paris with his journal detailing due to the absorption of light from such distances. So measurements, Arago was welcomed as a hero. He be- the measurement of some angles was postponed until came an assistant professor at L’École Polytechnique and a more favourable weather conditions with a clearer at- member of the Academy of Sciences, and he also worked mosphere prevailed. They had to wait for months in at the Paris Observatory, of which he became director in some locations. In March 1807, they both travelled to 1843. the island of Formentera, but their repeating circle According to their work on extending the Paris me- broke, so they had to return to mainland Spain. Biot ridian as far as Formentera, Arago and Biot determined then returned to Paris to have the instrument repaired, that one ten-millionth part of the Earth’s meridian from while Arago stayed in Spain and used other instruments the North Pole to the Equator was the equivalent of to repeat measurements along the Spanish coast. Arago 443.31 lignes (URL14). However, according to (URL14), was from the eastern part of the Pyrenees, so he spoke that result was amended in a later work to 443.39 lignes. Catalan, which helped him in his work, as well as pre- According to the reference ellipsoid WGS 84 (World venting him from being arrested. Geodetic System) determined using contemporary Biot returned to Spain with the repaired instrument satellite techniques, the modern value of one ten-milli- in November 1807 and joined Arago in Valencia. They onth part of meridian arc length equals 443.38308 lignes, travelled to the island of Formentera again, observed the i.e. 1.00019657 m. Therefore, in relation to the definition positions of stars and determined the latitude of the of the metre, the difference is 0.19657 mm. southernmost point of the Paris meridian. In addition, Arago was an imperturbable republican who parti- they determined the length of the seconds pendulum. cipated in politics, enabling the financing of new astro- Two Frenchmen working in Spanish territory with a sop- nomic instruments for the Paris Observatory and state histicated instrument raised suspicions, especially since support for the Academy of Sciences. the French had conquered Portugal and occupied part of It should be noted that the Archive Metre remained Spain. So in those tumultuous times, Biot returned to the standard metre in France in practice, even when it France and Arago continued working in Spain, disguised became known that it did not exactly correspond to the as a Spaniard. He went to Majorca, where he set up the new definition of the metre (URL 14). The introduction Clop de Galatzó (S'Esclop) trigonometric point and man- of new units of measure was very slow in France and aged to connect Majorca with Formentera, Ibiza, Mongo some other countries. Even today, the international SI and Desierto de las Palmas in a large trigonometric system is not accepted everywhere. The discovery that KiG No. 20, Vol. 1 2, 201 3 30 SOLARIĆ, M. I SOLARIĆ, N.: DULJINA LUKA PARIŠKOG MERIDIJANA I DEFINICIJA METRA

Također, na približno takvoj udaljenosti mogla se vidjeti i točka Campvey na otoku Ibizi. Iz te dvije točke mogla se odrediti i točka Galatzo (S'Esclop) na otoku Mallorci. Pritom je bila ostvarena i stranica Desierto de las Palmas – Campvey, duga više od 150 km (Murdin 2009, str. 113). Tako duge stranice u trokutima stvarale su poteškoće u mjerenju kutova zbog apsorpcije zraka svjetlosti koje dolaze s tako velikih udaljenosti. Zato su za ostvarenje mjerenja nekih kutova na udaljene točke morali čekati povoljnije vremenske prilike s prozračnom atmosferom. Na nekim mjestima morali su čekati mjesecima. U ožuj- ku 1807. obojica su putovali na otok Formentera, ali se njihov Bordin repeticijski krug slomio pa su se morali vratiti na kopneni dio Španjolske. Biot se tada vratio u Pariz noseći instrument na popravak, a Arago je ostao u Španjolskoj i upotrijebio preostale instrumente za po- navljanje mjerenja uzduž španjolske obale. Arago je po- tjecao iz istočnog dijela Pireneja, te je tako znao kata- lonski jezik što mu je znatno pomoglo u radu, a u nekim ga je slučajevima i spašavalo od zatočenja u zatvor. Biot se vratio u Španjolsku s popravljenim instru- Figure 1 0. Trigonometric network by Biot and Arago, mentom u studenome 1807. i pridružio se Aragou u Va- 1 806–1 809, extending the Paris meridian to the Balearic Islands lenciji. Tada su ponovno išli na otočić Formenteru, gdje (Natural Earth data and according to URL 1 3) su opažali položaje zvijezda i određivali geografsku širi- Slika 1 0. Trigonometrijska mreža Biota i Aragoa iz 1 806–1 809. nu te najjužnije točke na Pariškom meridijanu. Osim to- godine za produživanje Pariškog meridijana do Balearskih otoka ga određivali su i duljinu sekundnog njihala na tom (podloga Natural Earth i prema URL 1 3) mjestu. Dvojica Francuza sa sofisticiranim mjernim ins- trumentima koji rade na španjolskom teritoriju postali Opet se sudbina poigrala s Aragoom jer je nevrijeme su sumnjivi, posebice stoga što su Francuzi osvojili Por- bacilo brod na obalu Sjeverne Afrike, gdje su ga zarobili tugal i okupirali dio Španjolske. U tim burnim vremeni- muslimani. Zahvaljujući snalažljivosti stigao je 25. pro- ma Biot je pobjegao natrag u Francusku, a Arago je sinca u Alžir, odakle je uz pomoć francuskoga konzula nastavio raditi u Španjolskoj, prikriven kao Španjolac. otputovao po treći put u Marseille. Ovaj put je stigao u Tako se uputio i u Mallorcu, gdje je postavio trigonome- svoje odredište u Francusku bez nezgoda 2. srpnja 1809. trijsku točku Clop de Galatzó (S'Esclop) i uspio povezati Po dolasku u Pariz sa svojim dnevnikom u kojem su Mallorcu s Formenterom, Ibizom, Mongom i s Desierto bila zapisana i mjerenja, prihvaćen je kao heroj. Postao de las Palmasom u veliku trigonometrijsku mrežu (sl. je docent na École Polytechnique i član Akademije znanos- 10). Pritom je mjerio i geografske širine na nekoliko to- ti, a radio je i na Pariškom opservatoriju, gdje je postav- čaka (Murdin 2009, str. 116). ljen za ravnatelja 1843. godine. Kako su se Francuzi željeli proširiti i na Balearsko Arago i Biot su svojim mjerenjima na produživanju otočje, stanovništvo je osvjetljavanje vrha brda Galatzó Pariškog meridijana do otoka Formentera utvrdili da 10- (katalonski Mola de l'Esclop), koje je Aragou služilo za milijunti dio Zemljina meridijana od pola do ekvatora vidljivost te točke na veliku daljinu u trigonometrijskoj ima 443,31 akademijinu liniju (lignes) (URL14). Međutim, mreži, protumačilo kao aktivnost špijuna za potrebe prema (URL14) rečeno je da je u kasnijem radu taj rezul- invazije protivničke vojske. Stoga je Arago morao ići u tat povećan na 443,39 akademijinih linija. Moderna vri- zatvor u lipnju 1808., a 28. srpnja je pobjegao s otoka u jednost 10-milijuntog dijela duljine luka meridijana ribarskom brodu te 3. kolovoza stigao u Alžir. Dozvolu za prema referentnom elipsoidu WGS84 (World Geodetic putovanje u Marseille iz Alžira dobio je, ali kad se brod Sistem), određenom uz pomoć najsuvremenijih satelit- približio Marseilleu 16. kolovoza pao je kao zarobljenik u skih tehnika mjerenja iznosi 443,38308 akademijinih li- ruke Španjolaca. Posada i Arago bili su odvedeni kao za- nija, tj. 1,00019657 m. Dakle, razlika metra prema defi- robljenici u Roses na Španjolskoj obali, a poslije su ih niranoj zamisli iznosi 0,19657 mm. prebacili u Palamos. Međutim, Arago se ipak uspio 28. Arago je bio nepokolebljivi republikanac i sudjelovao studenoga 1808. ukrcati na brod za Marseille. je u političkom životu zemlje pa je na taj način omogućio KiG Br. 20, Vol. 1 2, 201 3 31 SOLARIĆ, M. AND SOLARIĆ, N.: THE PARIS MERIDIAN ARC LENGTH AND DEFINITION OF THE METRE the Archive Metre had shortcomings led to the produc- speed of light in a vacuum as equal to the distance light tion of premeasures for the metre and kilogram, com- travels in a vacuum during one 299 792 458th of a second posed of 90% platinum and 10% iridium, ensuring that no (Benčić 1984, URL14). The length of metre determ- changes would be made over a long period of time (Fig ined in such way is especially suitable for determin- 11). During its first assembly in 1889, the General Confer- ing distances in geodesy and astronomy, and can be ence on Weights and Measures proclaimed one of forty reproduced anywhere in the world. existing premeasures as the international premetre (A6) and defined the unit of length called the "metre" as an 4 Conclusion interval between two middle lines on the metre premeasure at 0 °C, preserved in the International Bureau of Weights and Meas- The survey of the Paris meridian helped define the ures in Sèvres. The premeasure is exactly 1 metre long if the at- length of the metre, although the length could have mospheric temperature is 0 °C and pressure is normal when it is been determined arbitrarily. It is necessary to point out supported in a horizontal position with two cylinders with dia- that the desire was to accept this measurement unit for meters of at least 1 cm which are 571 mm apart. the whole world and for all times. An attempt was made to define the metre as accur- ately as possible by applying new understandings in Acknowledgements physics, so that its length could be reproduced any- where on Earth as accurately as possible. Thus it was We would like to thank our reviewers for their help- changed in 1960 and 1983: ful comments, which contributed to the quality of this . The so-called wave metre came into effect from 1960, research on the history of geodesy. The survey of the without changing the length of the 1889 metre. It was Paris meridian helped define the metre as the basic unit defined as the length equal to 1 650 763.73 of wavelength in of length in the international system. vacuum of radiation corresponding to the transition We would also like to thank the Ministry of Science, betweenlevels2p10and5d3ofthecrypton86atom.Itcould Education and Sport of the Republic of Croatia for par- be reproduced anywhere in the world with a reliabil- tially financing this paper, which was written as part of ity of about ±10–8 m = ±0.01 μm (Brezinšćak 1971). the project Development of Scientific Measuring Laboratory . Since 1983, the metre has been defined using the for Geodetic Instruments, No. 007-1201785-3539.

References / Literatura

Benčić, D. (1 984): O osnovnoj mjernoj jedinici u geodeziji, Geodetski list br. 1 –3, str. 5–1 1 Bialas, V. (1 982): Erdgestalt, Kosmologie und Weltanschauung, Konrad Wittwer, Stuttgart. Brezinšćak, M. (1 971 ): Mjerenje i računanje u tehnici i znanosti, Tehnička knjiga, Zagreb. Murdin, P. (2009): Full Meridian of Glory, Springer Science + Business Media, USA. Solarić, M., Solarić, N. (201 3): Francuska geodetska znanstvena ekspedicija u Peru (in preparation). Vykutil, J. (1 982): Vyšší geodézie, Kartografie, Praha.

URLs / Mrežne adrese

URL 1 : Jean Baptiste Joseph Delambre, http://www-history.mcs.st-and.ac.uk/Mathematicians/Delambre.html (6. 7. 201 1 .) URL 2: Jean Baptiste Joseph Delambre, http://en.wikipedia.org/wiki/Jean_Baptiste_Joseph_Delambre (7. 3. 201 1 .) URL 3: Pierre Méchain, http://www-history.mcs.st-and.ac.uk/Biographies/Mechain.html (6. 7. 201 1 .) URL 4: Pierre Méchain, http://en.wikipedia.org/wiki/Pierre_M%C3%A9chain (7. 3. 201 1 .) URL 5: The history of measurement, http://www-history.mcs.st-and.ac.uk/HistTopics/Measurement.html (7. 7. 201 1 .) URL 6: Figure de la Terre et méridienne de Delambre et Méchain, http://fr.wikipedia.org/ (1 2. 7. 201 1 .) URL 7: Francois Arago, http://fr.wikipedia.org/wiki/Fran%C3%A7ois_Arago (1 2. 7. 201 1 .) URL 8: Montjuic Castle, http://en.wikipedia.org/wiki/Montju%C3%AFc_Castle (1 2. 7. 201 1 .) URL 9: Jean-Baptiste Biot, http://en.wikipedia.org/wiki/Jean-Baptiste_Biot (1 2. 7. 201 1 .) URL 1 0: Jean-Baptiste Biot, http://www-history.mcs.st-and.ac.uk/Biographies/Biot.html (1 . 8. 201 1 .) URL 1 1 : Cartography and the Cassini and Maraldi families, http://www.catnaps.org/cassini/cart.html (28. 3. 201 1 ) URL 1 2: Dominique François Jean Arago, http://www-history.mcs.st-and.ac.uk/Biographies/Arago.html (7. 8. 201 1 .) URL 1 3: In the Foot steps of Arago, http://lec.ugr.es/~julyan/papers/arago/indeks.html (31 . 8. 201 1 .) URL 1 4: History of metre, http://en.wikipedia.org/wiki/History_of_the_metre (31 . 8. 201 1 .) URL 1 5: Cathedral in Rodez, http://en.wikipedia.org/wiki/Rodez_Cathedral (2. 9. 201 1 .) URL 1 6: Un historique du metre, http://www.industrie.gouv.fr/metro/aquoisert/metre.htm (20. 7. 201 1 .)

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Figure 1 1 . Diagrams showing the dimensions of the international historical metre with lines engraved in a neutral plane. All measurements are in mm. Slika 1 1 . Grafički prikaz dimenzija međunarodnog prametra s crticama ugraviranim u neutralnu ravninu. Mjere su izražene u milimetrima financiranje kupnje novih astronomskih instrumenata Metar je duljina jednaka 1 650 763,73 duljine vala u zrako- za Pariški opservatorij, a osigurao je i financijsku potpo- praznom prostoru zračenja koje odgovara prijelazu između ru države Akademiji znanosti. razina 2 p10 i 5 d3 atoma kriptona 86. Treba primijetiti da je Arhivski metar ostao pravno Mogao se reproducirati bilo gdje na svijetu s pouzda- praktički standard za metar u Francuskoj čak i onda ka- nošću oko ± 10–8 m = ± 0,01 μm (Brezinšćak 1971). da je bilo poznato da ne odgovara točno definiciji metra . Od 1983. godine metar je definiran s pomoću brzine (URL14). Uvođenje novih mjernih jedinica u život išlo je svjetlosti u zrakopraznom prostoru ovako: vrlo sporo u Francuskoj, ali i u drugim zemljama tako da Metar je jednak duljini puta koji svjetlost prijeđe u zrako- i danas neke zemlje nisu prihvatile taj međunarodni praznom prostoru za vrijeme 299 792 458-og dijela sekunde sustav mjernih jedinica (System International, skraćeno SI (Benčić 1984, URL14). sustav). Poslije je utvrđeno da postoje i neki nedostaci Tako definiran metar posebno je pogodan za određi- Arhivskog metra. Zato se studiozno pristupilo izradi Pra- vanje udaljenosti u geodeziji i astronomiji, a može se mjera metra i kilograma izrađenih od 90% platine i 10% iri- reproducirati bilo gdje na svijetu. dija, a pritom se posebno pazilo da u dugom razdoblju ne dođe do bilo kakvih promjena (sl. 11). Generalna konfe- 4. Zaključak rencija za mjere i utege na svom prvom zasjedanju 1889. godine proglasila je jedan od četrdesetak izvanrednih Izmjera Pariškog meridijana pomogla je određivanju primjeraka prametra međunarodnim prametrom (obi- duljine metra, iako se mora priznati da se ta duljina mo- lježen oznakom A6) i jedinicu duljine ''metar'' definirala: gla uzeti proizvoljno. Međutim, treba naglasiti da se že- Jedinica duljine je metar koji je pri temperaturi 0 °C definiran ljelo da ta mjerna jedinica bude prihvatljiva za sve razmakom između dvije srednje crtice na pramjeri metra, po- narode svijeta i za sva vremena. hranjenog u Međunarodnom uredu za mjere i utege u Sèvresu. Pramjera ima točno duljinu 1 metar kad je pri temperaturi 0 °C Zahvala i normalnom atmosferskom tlaku poduprta u vodoravnom po- ložaju s dva valjka promjera najmanje 1 cm, koji su međusobno Najljepše zahvaljujemo recenzentima na korisnim udaljeni 571 mm. primjedbama, kojima su pridonijeli boljoj kvaliteti ovog Tijekom vremena nastojala se što točnije definirati istraživanja geodetske prošlosti. Izmjera Pariškog meri- duljina metra s novijim spoznajama iz fizike tako da bi se dijana pripomogla je definiciji metra kao osnovne jedi- mogla reproducirati duljina metra na bilo kojem mjestu nice za duljinu u međunarodnom sustavu jedinica. na Zemlji i što točnije, te su one mijenjane 1960. i 1983. Zahvaljujemo također Ministarstvu znanosti, obra- godine: zovanja i sporta RH, što je djelomično financiralo ovaj . Od 1960. godine bio je službeno na snazi tzv. valni me- rad koji je izrađen u okviru projekta Razvoj znanstvenog tar, a pritom nije mijenjana duljina metra iz 1889. godi- mjeriteljskog laboratorija za geodetske instrumente br. 007- ne. On je definiran kao: 1201785-3539.

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